Hira L. Koul

Weighted Empirical Processes in Dynamic Nonlinear Models

• Broschiertes Buch

Produktbeschreibung

This book presents a unified approach for obtaining the limiting distributions of minimum distance, M and R estimators corresponding to non-smooth underlying scores in a large class of dynamic non-linear models including ARCH models. It also discusses classes of goodness-of-t tests for fitting an error distribution in some of these models and/or fitting a regression-autoregressive function without assuming the knowledge of the error distribution. The main tool is the asymptotic equicontinuity of certain basic weighted residual empirical processes in the uniform and L2 metrics. The contents of this monograph should be useful to graduate students and research scholars in statistics, econometrics, and finance. This book is a an updated edition of the author's monograph "Weighted Empirical Processes and Linear Models" (IMS Lecture Notes-Monograph 21, 1992). The new edition differs from the previous one in many ways. To mention just a few: It includes asymptotically distribution free tests for fitting a regression and/or an autoregressive models; the asymptotic distributions of auto-regression quantiles and rank scores; and above all the weak convergence of the residual empirical processes useful in nonlinear ARCH models. Hira L. Koul is a professor of statistics at Michigan State University. He is a Fellow of the IMS and an Elected Member of the International Statistical Institute. He was awarded the prestigious Humboldt Research Award for Senior Researchers in 1995. He has been on the editorial boards of the Annals of Statistics, Sankhya, and J. Indian Statistical Association. Currently he is a Coordinating Editor of the Journal of Statistical Planning and Inference, and an Associate Editor of Statistics and Probability Letters.

---

The role of the weak convergence technique via weighted empirical processes has proved to be very useful in advancing the development of the asymptotic theory of the so called robust inference procedures corresponding to non-smooth score functions from linear models to nonlinear dynamic models in the 1990's. This monograph is an ex panded version of the monograph Weighted Empiricals and Linear Models, IMS Lecture Notes-Monograph, 21 published in 1992, that includes some aspects of this development. The new inclusions are as follows. Theorems 2. 2. 4 and 2. 2. 5 give an extension of the Theorem 2. 2. 3 (old Theorem 2. 2b. 1) to the unbounded random weights case. These results are found useful in Chapters 7 and 8 when dealing with ho moscedastic and conditionally heteroscedastic autoregressive models, actively researched family of dynamic models in time series analysis in the 1990's. The weak convergence results pertaining to the partial sum process given in Theorems 2. 2. 6 . and 2. 2. 7 are found useful in fitting a parametric autoregressive model as is expounded in Section 7. 7 in some detail. Section 6. 6 discusses the related problem of fit ting a regression model, using a certain partial sum process. Inboth sections a certain transform of the underlying process is shown to provide asymptotically distribution free tests. Other important changes are as follows. Theorem 7. 3.

Produktdetails

• Lecture Notes in Statistics 166
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 978-0-387-95476-9
• 2nd ed.
• Seitenzahl: 448
• Erscheinungstermin: 13. Juni 2002
• Englisch
• Abmessung: 235mm x 155mm x 25mm
• Gewicht: 636g
• ISBN-13: 9780387954769
• ISBN-10: 0387954767
• Artikelnr.: 10904923

Autorenporträt

Hira L. Koul is a professor of statistics at Michigan State University. He is a Fellow of the IMS and an Elected Member of the International Statistical Institute. He was awarded the presti- gious Humboldt Research Award for Senior Researchers in 1995. He has been on the editorial boards of the Annals of Statistics, Sankhya, and J. Indian Statistical Association. Currently he is a Coordinating Editor of the Journal of Statistical Planning and Inference, and an Associate Editor of Statistics and Probability Letters.

Inhaltsangabe

1 Introduction.- 1.1 Weighted Empirical Processes.- 1.2 M-, R- and Scale Estimators.- 1.3 M.D. Estimators & Goodness-of-fit Tests.- 1.4 R.W.E. Processes and Dynamic Models.- 2 Asymptotic Properties of W.E.P.'s.- 2.1 Introduction.- 2.2 Weak Convergence.- 2.3 AUL of Residual W.E.P.'s.- 2.4 Some Additional Results for W.E.P.'S.- 3 Linear Rank and Signed Rank Statistics 69.- 3.1 Introduction.- 3.2 AUL of Lin ear Rank Statistics.- 3.3 AUL of Linear Signed Rank Statistics.- 3.4 Weak Convergence of Rank and Signed Rank W.E.P.'s.- 4 M, R and Some Scale Estimators.- 4.1 Introduction.- 4.2 M-Estimators.- 4.3 Distributions of Some Scale Estimators.- 4.4 R-Estimators.- 4.5 Est imation of Q(f).- 5 Minimum Distance Estimators.- 5.1 Introduction.- 5.2 Definitions of M.D. Estimators.- 5.3 Finite Sample Properties.- 5.4 A General M.D. Estimator.- 5.5 Asymptotic Uniform Quadraticity.- 5.6 Distributions, Efficiency & Robustness.- 6 Goodness-of-fit Tests in Regression.- 6.1 Introducti on.- 6.2 The Supremum Distance Tests.- 6.3 L2-Distance Tests.- 6.4 Testing with Unknown Scale.- 6.5 Testing for the Symmetry of the Errors.- 6.6 Regression Model Fitting.- 7 Autoregression.- 7.1 Introduction.- 7.2 AUL of Wh and Fn.- 7.3 GM- and GR- Estimators.- 7.4 Minimum Distance Estimation.- 7.5 Autoregression Quantiles and Rank Scores.- 7.6 Goodness-of-fit Testing for F.- 7.7 Autoregressive Model Fitting.- 8 Nonlinear Autoregression 358.- 8.1 Introduction.- 8.2 AR Models.- 8.3 ARCH Models.- Lectures Notes in Statistics.